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Healthy and cancer tissue growth, therapies and the
use of mathematical [cell population dynamic] models
Jean Clairambault
INRIA, Bang project-team, Paris & Rocquencourt, France
Laboratoire Jacques-Louis Lions, UPMC, Paris, France
INSERM U776, « Biologic rhythms and cancers», Villejuif, France
http://www.rocq.inria.fr/bang/JC/
[email protected]
Motivation: modelling tissue growth to represent proliferating
cell populations as targets, wanted or unwanted, of anticancer
drugs, to optimise their delivery at the whole body level.
Main pitfalls of cancer therapeutics in the clinic:
- unwanted toxic side effects on healthy cell populations
- emergence of drug-resistant clones in cancer cell populations
Designing a model of tissue growth must be undertaken after
answering the question: Modelling, what for? Resulting models
may be very different according to the answers to this question.
Recalling a few facts about healthy and cancer tissue growth
Tissues that may evolve toward malignancy
…are the tissues where cells are committed to fast proliferation
(fast renewing tissues):
-epithelial cells+++, i.e., cells belonging to those tissues which
cover the free surfaces of the body (i.e., epithelia):
gut (colorectal cancer), lung, glandular coverings (breast, prostate),skin,…
-cells belonging to the different blood lineages, produced in
the bone marrow: liquid tumours, or malignant haemopathies
-others (rare: sarcomas, neuroblastomas, dysembryomas…)
Natural history of cancers: from genes to bedside
Gene mutations: an evolutionary process which may give rise to abnormal DNA
when a cell duplicates its genome, due to defects in tumour suppressor or DNA
mismatch repair genes (Yashiro et al. Canc Res. 2001; Gatenby & Vincent, Canc. Res. 2003)
Resulting genomic instability allows malignant cells to escape proliferation and
growth control at different levels: subcellular, cell, tissue and whole organism:
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Entry in the cell cycle for quiescent (=non-proliferating) cells
Phase transitions and apoptosis for cycling cells
Ability to use anaerobic glycolysis (metabolic selective advantage for cancer cells)
Contact inhibition by surrounding cells (cell adhesion, cell density pressure)
Ability to stimulate formation of new blood vessels from the neighbourhood
Linking to the extracellular matrix (ECM) fibre network and basal membranes
Recognition (friend or foe) by the immune system
Cancer invasion is the macroscopic result of breaches in these control mechanisms
Evading proliferation and growth control mechanisms
Hanahan & Weinberg, Cell 2000
Hanahan & Weinberg, Cell 2011
…but just what is cell proliferation?
Cell population growth in proliferating tissues
(after Lodish et al., Molecular cell biology, Nov. 2003)
One cell divides in two: a controlled process at cell and tissue levels
... that is disrupted in cancer
Populations of cells:
the right observation level to model proliferation, apoptosis, differentiation
At the origin of proliferation: the cell division cycle
S:=DNA synthesis; G1,G2:=Gap1,2; M:=mitosis
Cyclin B
+ CDK1
G2
S
Cyclin D
+ CDK4, 6
M
G1
Cyclin A
+ CDK2
Cyclin E
+ CDK2
(after F. Lévi, INSERM U776)
(after Lodish et al., Molecular cell biology, Nov. 2003)
Physiological or therapeutic control on:
- transitions between phases
(G1/S, G2/M, M/G1)
- death rates (apoptosis or necrosis)
inside phases
- progression speed in phases (G1)
- exchanges with quiescent phase G0
Proliferating and quiescent cells
after R:
mitogen-independent
progression through G1 to S
(no way back to G0)
Restriction point
(in late G1 phase)
R
before R:
mitogen-dependent
progression through G1
(possible regression to G0)
From Vermeulen et al. Cell Prolif. 2003
Most cells do not proliferate physiologically, even in fast renewing tissues (e.g., gut)
Exchanges between proliferating (G1SG2M) and quiescent (G0) cell compartments
are controlled by mitogens and antimitogenic factors in G1 phase
Phase transitions, apoptosis and DNA mismatch repair
-Sensor proteins, e.g. p53, detect defects
in DNA, arrest the cycle at G1/S and
G2/M phase transitions to repair
damaged fragments, or lead the whole
cell toward controlled death = apoptosis
p53
Repair or apoptosis
-p53 expression is known to be downregulated in about 50% of cancers
-Physiological inputs, such as circadian
gene PER2, control p53 expression;
circadian clock disruptions (shiftwork)
may result in low p53-induced genomic
instability and higher incidence of cancer
M
G2
G1
S
p53
Repair or apoptosis
(F. Lévi, INSERM U776)
(Fu & Lee, Nature Rev. 2003)
Invasion: local, regional and remote
1) Local invasion by tumour cells implies loss of
normal cell-cell and cell-ECM (extracellular matrix)
contact inhibition of size growth and progression in the
cell cycle. ECM (fibronectin) is digested by tumoursecreted matrix degrading enzymes (MDE=PA, MMP)
so that tumour cells can move out of it. Until 106 cells
(1 mm ) is the tumour in the avascular stage.
2) To overcome the limitations of the avascular stage,
local tumour growth is enhanced by tumour-secreted
endothelial growth factors which call for new blood
vessel sprouts to bring nutrients and oxygen to the
insatiable tumour cells (angiogenesis, vasculogenesis)
3) Moving cancer cells can achieve intravasation, i.e.,
migration in blood and lymph vessels (by diapedesis),
and extravasation, i.e. evasion from vessels, through
vascular walls, to form new colonies in distant tissues.
These new colonies are metastases.
(Images thanks to A. Anderson, M. Chaplain, J. Sherratt, and Cl. Verdier)
Proliferating rim
Quiescent layer
Necrotic core
Mathematical models
(A reference: Optimisation of cancer drug treatments using cell population dynamics,
by F. Billy, JC, O. Fercoq, to appear soon in “Mathematical Models and Methods in
Biomedicine”, A. Friedman, E. Kashdan, U. Ledzewicz, H. Schättler eds. Springer, NY)
Mathematical models of healthy and tumour growth
(a great variety of models)
• In vivo (tumours) or in vitro (cultured cell colonies) growth? In vivo (diffusion in
living organisms) or in vitro (constant concentrations) growth control by drugs?
• Scale of description for the phenomenon of interest: subcellular, cell, tissue or whole
organism level? … may depend upon therapeutic description level
• Is space a relevant variable? [Not necessarily!] Must the cell cycle be represented?
• Are there surrounding tissue spatial limitations? Limitations by nutrient supply or
other metabolic factors?
• Is loco-regional invasion the main point? Then reaction-diffusion equations (e.g.
KPP-Fisher) are widely used, for instance to describe tumour propagation fronts
• Is cell migration to be considered? Then chemotaxis [=chemically induced cell
movement] models (e.g. Keller-Segel) have been used
Simple models of tumour growth, to begin with
Macroscopic, non-mechanistic models: the simplest ones:
exponential, logistic, Gompertz
x= tumour weight
or volume, proportional
to the number of cells,
or tumour cell density
x
t
Exponential model: relevant for the early stages of tumour growth only
[Logistic and] Gompertz model: represent growth limitations (S-shaped curves with
plateau=maximal growth, or tumour carrying capacity xmax), due to mechanical
pressure or nutrient scarcity
Models of tumour growth: Gompertz revisited
ODE models a) with 2 cell compartments, proliferating and quiescent,
or b) varying the tumour carrying capacity xmax in the original Gompertz model
(Gyllenberg & Webb, Growth, Dev. & Aging 1989; Kozusko & Bajzer, Math BioSci 2003)
Avowed aim: to mimic Gompertz growth (a popular model among radiologists)
However, a lot of cell colonies and tumours do not follow Gompertz growth
(Recent refinements: Hahnfeldt et al., Canc. Res 1999; Ergun et al., Bull Math Biol 2003)
Tumour burden
Example: tumour growth
(GOS) in a population of
mice, laboratory data
d14
d12
d9
Data
Gompertz model
d8
time
ODE models with two exchanging cell compartments,
proliferating (P) and quiescent (Q)
Cell exchanges
(Gyllenberg & Webb, Growth, Dev. & Aging 1989; Kozusko & Bajzer, Math BioSci 2003)
where, for instance:
r0 representing here the rate of
inactivation of proliferating cells,
and ri the rate of recruitment from
quiescence to proliferation
Initial goal: to mimic Gompertz growth
Models of tumour growth: stochastic
Physical laws describing macroscopic spatial dynamics of an avascular tumour
-Fractal-based phenomenological description of growth of cell colonies and tumours,
relying on observations and measures: roughness parameters for the 2D or 3D tumour
Findings: - all proliferation occurs at the outer rim
- cell diffusion along (not from) the tumour border or surface
- linear growth of the tumour radius after a critical time (before: exponential)
(A. Bru et al. Phys Rev Lett 1998, Biophys J 2003)
-Individual-based models (IBMs):
- cell division and motion described by
a stochastic algorithm, then continuous limit
- permanent regime = KPP-Fisher-like
(also linear growth of the tumour radius)
(D. Drasdo, Math Comp Modelling 2003; Phys Biol 2005
H. Byrne & D. Drasdo, J. Math. Biol. 2009)
Models of tumour growth: invasion
Macroscopic reaction-diffusion evolution equations
1 variable c = density of tumour cells): KPP-Fisher equation
D(x) = diffusion (motility) in brain tissue,
(reaction)=growth of tumour cells,1D x
spatial variable and c: density of tumour
cells, used to represent brain tumour radial
propagation from a centre
(K. Swanson & J. Murray, Cell Prolif 2000;
Br J Cancer 2002; J Neurol Sci 2003)
Models of tumour growth: invasion as competition
Macroscopic reaction-diffusion equations to represent invasion (recession) fronts
2 or more variables: ex.: healthy cells N1, tumour cells N2, excess H+ ions L
N2
N1
L
Prediction: interstitial cell gap between tumour
propagation and healthy tissue recession fronts
(Gatenby & Gawlinski, Canc. Res. 1996)
Models for moving tumour cells in the ECM
Chemotaxis: chemo-attractant induced cell movements
Keller-Segel model
p = density of cells
w = density of chemical
(Originally designed for movements of bacteria, with w=[cAMP])
(Keller & Segel, J Theoret Biol 1971)
Anderson-Chaplain model for local invasion by tumour cells in the ECM
n = density of cells
f = ECM density
m = MDE (tumour
metalloproteases)
u = MDE inhibitor
(Anderson & Chaplain, Chap 10 in Cancer modelling and simulation, L. Preziosi Ed, Chapman & Hall 2003)
Models for angiogenesis
VEGF-induced endothelial cell movements towards tumour
- Biochemical enzyme kinetics
- Chemical transport (capillary and ECM)
- “Reinforced random walks”
- Cell movements in the ECM
Models by Anderson and Chaplain,
Levine and Sleeman
(Levine & Sleeman,Chap. 6 in Cancer modelling and
simulation, L. Preziosi Ed, Chapman & Hall 2003)
A multiscale angiogenesis model
Interacting cell populations
Proliferating cancer cell population
Coupling by oxygen concentration, acting on actual commitment of cells
into the division cycle (passing the restriction point)
Aim: assessment of an antiangiogenic treatment by endostatin
F. Billy et al., J. Theor. Biol. 2009
Modelling the cell cycle 1: single cell level
Ordinary differential equations to describe progression in the cell cycle
A. Golbeter’s minimal model for the « mitotic oscillator », i.e., G2/M phase transition
C
M
X
C = cyclin B, M = Cyclin-linked cyclin dependent kinase, X = anticyclin protease
Switch-like dynamics of kinase cdk1, M
Adapted to describe G2/M phase transition,
which is controlled by Cyclin B-CDK1
(A. Goldebeter, Biochemical oscillations and cellular rhythms, CUP 1996)
Modelling the cell cycle 2: single cell (continued)
Detailed ODE models to describe progression in the cell cycle: Tyson & Novak
Phase transitions:
-G1/S
-G2/M
-Metaphase/anaphase
…due to steep variations
of Cyc-CDK concentrations
(bistability with hysteresis)
(Novak, Bioinformatics 1999)
(Tyson, Chen, Novak, Nature Reviews 2001)
Modelling the cell cycle 2: single cell (continued)
Detailed ODE models to describe progression in the cell cycle: Gérard & Goldbeter
39 variables. Growth factor, rather than cell mass
(as in Tyson & Novak) is the driving parameter
A simplified model has been proposed, with 5 variables
C. Gérard & A. Goldbeter, PNAS 2009; Interface Focus 2011
C. Gérard, D. Gonze & A. Goldbeter, FEBS Journal 2012
Modelling the cell cycle 3: cell population level
Age-structured PDE models for proliferating cell populations
(after B. Basse et al., J Math Biol 2003)
In each phase i , a Von Foerster-McKendrick-like equation:
Flow cytometry may help quantify
proliferating cell population repartition
according to cell cycle phases
ni:=cell population
density in phase i
di:=death rate
K i->i+1:=transition rate
(with a factor 2for i=1)
di , K i->i+1 constant or
periodic w. r. to time t
(1≤i≤I, I+1=1)
Death rates di and phase transitions K i->i+1 are targets
for physiological (e.g. circadian) and therapeutic (drugs) control
: a growth exponent governing the cell population behaviour
Otherwise said: proof of the existence of a unique growth exponent , the same for
all phases i, such that the
are bounded, and asymptotically
periodic if the control is periodic.
Example (periodic control case): 2 phases, control on G2/M by 24 h-entrained periodic
CDK1-Cyclin B (from A. Goldbeter’s minimal mitotic oscillator model)
“Surfing on the
exponential growth
curve”
time t
=CDK1-Cyc B
Cells in G1-S-G2 (phase i=1) Cells in M (phase i=2)
Entrainment of the cell cycle by = CDK1-Cyc B at a 24 h-period
(= the same as adding
an artificial death term
 to the di, stabilising
the system)
Details (1): 2-phase model, no control on transition
Asynchronous cell growth
The total population of cells
inside each phase follows
asymptotically an exponential
behaviour
Stationary state
distribution of cells
inside phases
according to age a:
no control ->
exponential decay
Details (2): 2 phases, periodic control  on G2/M transition
Synchronisation between cell
cycle phases at transitions
The total population of cells
inside each phase follows
asymptotically an exponential
behaviour tuned by a periodic
function
Stationary state
distribution of cells
inside phases
according to age a:
sharp periodic
control ->sharp rise
and decay
The simplest case: 1-phase model with division
(Here, v(a)=1, a* is the cell cycle duration, and is the time
during which the 1-periodic control  is actually exerted on cell division)
Then it can be shown that the eigenvalue problem:
has a unique positive 1-periodic eigenvector N, with a positive eigenvalue 
Experimental measurements to identify transition kernels Ki_i+1
(and simultaneously experimental evaluation of the first eigenvalue )
In the simplest model with d=0 (one phase, no cell death) and assuming K=K(x)
(instead of indicator functions
, experimentally more realistic transitions):
Whence (by integration
along characteristic lines):
Interpreted as: if  is the age in phase at division, or transition:
with
With probability density (experimentally identifiable):
i.e.,
Experimental identification of basic model parameters
with FUCCI reporters on a 2-phase model G1 / S-G2-M
FUCCI=Fluorescent Ubiquitination-based Cell Cycle Indicator
FUCCI: a movie (Sakaue-Sawano 2008) on HeLa cells
Another FUCCI movie (C. Feillet, IBDC Nice), NIH3T3 cells
FUCCI reporters
Measuring time intervals: G1 and total division cycle durations
Data from G. van der Horst’s lab, Erasmus University, Rotterdam, processed by Frédérique Billy at INRIA
(NIH3T3 cells)
(Billy et al., Math. Comp. Simul. 2012)
Phase transitions w.r.t. age x
Gamma p.d.f.s f(x) fitted from data on 55 NIH 3T3 cells
(Billy et al., Math. Comp. Simul. 2012)
Phase transitions w.r.t. age x
Transition rates K(x) from p.d.f.s f(x) on NIH 3T3 cells
and resulting population evolution on one complete cycle
K1 (G1 to S)
K2 (M to G1)
G1
SG2M
(Billy et al., Math. Comp. Simul. 2012)
A possible application to the investigation of
synchronisation between cell cycle phases
(from Lodish et al., Molecular cell biology, Nov. 2003)
One cell divides in two: a physiologically controlled process at cell and tissue levels
in all healthy and fast renewing tissues (gut, bone marrow) that is disrupted in cancer:
Is cell cycle phase synchronisation a mark of health in tissues?
A working hypothesis that could explain differences in
responses to drug treatments between healthy and cancer tissues
Healthy tissues, i.e., cell populations, would be well synchronised
w. r. to proliferation rhythms and w. r. to circadian clocks, whereas…
...tumour cell populations would be desynchronised w. r. to both, and such
proliferation desynchronisation would be a consequence of an escape
by tumour cells from central circadian clock control messages, just as
they evade most physiological controls, cf. e.g., Hanahan & Weinberg:
Question:
is cell cycle phase
desynchronisation
another hallmark of
cancer in cell
populations?
A mathematical result:  increases with desynchronisation
(expected, and shown in Th. Ouillon’s internship report, INRIA 2010)
where desynchronisation is defined as a measure of phase overlapping at transition
i.e.,for a given family (fi) of p.d.f.s with second moment si,  is increasing with each si
(also shown in Billy et al., Math. Comp. Simul., 2012)
Computing the growth exponent, fitting data to p.d.f.s:
Gamma p.d.f.s were best fits and yielded simple computations:
2-phase Lotka’s equation simply reads:
... which yields here  = 0.039 h-1
(and yields mean doubling time Td =17.77 h, and mean cell cycle time Tc =17.95 h)
Now, what if we add periodic (e.g., circadian) control??
i.e., what if we put K(x,t) = k(x).(t)
with k = FUCCI-identified and  = a cosine?
[cosine: in the absence of a better identified clock thus far]
Phases: asynchronous cell growth
Global: sheer exponential cell growth
[Agreement between
model and data on
the first division]
F. Billy
Steep synchronisation within the cell cycle
Stepwise cell population growth
F. Bill
Soft synchronisation within the cell cycle
Stepwise cell population growth
F. Billy
F. Billy
Other cell population dynamic models relying on the cell cycle
A model of tissue growth with proliferation/quiescence
An age[a]-and-cyclin[x]-structured PDE model with proliferating and quiescent cells
(exchanges between (p) and (q), healthy and tumour tissue cases: G0 to G1 recruitment differs)
p: proliferating
cells
q: quiescent cells
N: all cells (p+q)
Healthy tissue
recruitment:
homeostasis
0
for small 
0
for large N
Tumour recruitment:
exponential (possibly
polynomial) growth
F. Bekkal Brikci,
JC, B. Ribba,
B. Perthame
JMB 2008; MCM
2008
0
for all 
M. DoumicJauffret, MMNP
2008
P-Q again: Simple PDE models, age-structured with
exchanges between proliferation and quiescence
p=density of proliferating cells; q=density of quiescent cells; ,=death terms;
K=term describing cells leaving proliferation to quiescence, due to mitosis;
=term describing “reintroduction” (or recruitment) from quiescence to proliferation
Delay differential models with two cell compartments,
proliferating (P)/quiescent (Q): Haematopoiesis models
(obtained from the previous model with additional hypotheses and integration in age x along characteristics)
(delay  cell division cycle time)
(from Mackey, Blood 1978)
Properties of this model: depending on the parameters, one can have positive
stability, extinction, explosion, or sustained oscillations of both populations
(Hayes stability criteria, see Hayes, J London Math Soc 1950)
Oscillatory behaviour is observed in periodic Chronic Myelogenous Leukaemia
(CML) where oscillations with limited amplitude are compatible with survival,
whereas explosion (blast crisis, alias acutisation) leads to AML and death
(Mackey and Bélair in Montréal; Adimy, Bernard, Crauste, Pujo-Menjouet, Volpert in Lyon)
From Adimy, Crauste, ElAbdllaoui J Biol Syst 2008 (see also: Özbay, Bonnet, Benjelloun, JC MMNP 2012)
Modelling haematopoiesis
for Acute Myelogenous Leukaemia (AML)
…aiming at non-cell-killing therapeutics
by inducing re-differentiation of cells using
molecules (e.g. ATRA) enhancing differentiation
rates represented by Ki terms
where ri and pi represent resting and proliferating
cells, respectively, with reintroduction term i=i(xi)
positive decaying to zero,
with population argument:
and boundary conditions:
Modelling leukaemic haematopoiesis:
(Mackey/Adimy) proliferation advantage?
TK (flt3-ITD) mutation
Stem cells CD34+/CD38-
Committed cells CD34+/CD38+
Blood/ bone marrow sampling
in AML patients
Cell sorting (magnetic beads)
FACS for cell cycle phases
Self-renewal
Measuring apoptosis and cell
division in each population
Model adaptation (in charge: A. Ballesta, Hosp. St Antoine)
Optimisation of cancer therapy by cytotoxic drugs
• Optimal control strategies to overcome the development of drug resistant cell
populations, using different drugs (Kimmel & Swierniak, 2006)
• Pulsed chemotherapies aiming at synchronising drug injections with cell cycle
events to enhance the effect of drugs on tumours: e.g. optimal control of IL21
injection times and doses  ui  (t-ti) using variational methods (Z. Agur, IMBM, Israel)
•
Chronotherapy = continuous infusion time regimens taking advantage
of optimal circadian anti-tumour efficacy and healthy tissue tolerability
for each particular drug: has been in use for the last 15 years, with particular
achievements for colorectal cancer (F. Lévi, INSERM U776, M.-C. Mormont & F. Lévi, Cancer 2003)
Examples of (still theoretical) applications to therapeutics:
Controlling unwanted toxic side effects on healthy tissues
PK-PD simplified model for cancer chronotherapy
(here with only toxicity constraints; target=death rate)
Tumour cells (Glasgow osteosarcoma)
Healthy cells (jejunal mucosa)
(PK)
(pop.
dyn.)
(homeostasis=damped harmonic oscillator)
(tumour growth=Gompertz model)
(« chrono-PD »)
f(C,t)=F.C/(C50+C).{1+cos 2(t-S)/T}
g(D,t)=H.D/(D50+D).{1+cos 2(t-T)/T}
Aim: balancing IV delivered drug anti-tumour efficacy by healthy tissue toxicity
(JC et al. Pathol-Biol 2003; Basdevant, JC, Lévi M2AN 2005; JC ADDR 2007)
Optimal control, step 1: deriving an objective
function from the tumoral cell population model
 characteristic
function of the
allowed infusion
periods)
Eradication strategy: minimise GB(i), where:
or else:
Stabilisation strategy: minimise GB(i), where:
(t1 < tf being some fixed observation time
after t0,beginning of infusion interval)
Optimal control, step 2: deriving a constraint
function from the enterocyte population model
Minimal toxicity constraint, for 0<A<1 (e.g. A =50%):
±other possible constraints:
Optimal control problem: defining a lagrangian:
then:
If GB and FA were convex, then a necessary and sufficient condition would be:
…i.e. the minimum would be obtained at a saddle point of the lagrangian,
reachable by a Uzawa-like algorithm
Optimised control: results of a tumour stabilisation strategy
using this simple one-drug PK-PD model
Objective: minimising the maximum
of the tumour cell population
Constraint : preserving the jejunal mucosa
according to the patient’s state of health
Result : optimal infusion flow i(t) adaptable to the patient’s state of health
(according to a tunable parameter A: here preserving A=50% of enterocytes)
(Basdevant, JC, Lévi M2AN 2005; JC ADDR 2007; Lévi, Okyar, Dulong, JC Annu Rev Pharm Toxicol 2010)
Another way to take into account a toxicity constraint (with Frédérique Billy and Olivier Fercoq):
Theoretical chronotherapeutic optimisation
of a 1st eigenvalue (cancer growth) under the constraint
of preserving another 1st eigenvalue (healthy tissue growth)
- McKendrick’s model of cell population proliferation
- Control of proliferation by blocking Ki_i+1 using theoretic periodic drug delivery:
K(t,x)=[1-g(t)].(t).k(x) where: g(t) is a periodic external control (chronotherapy)
(t) is a circadian clock control on the cell cycle
k(x) is an [only] age-dependent transition rate
- Objective function to be minimised: , 1st eigenvalue of cancer cell population
- Constraint function to be preserved: 2  , 1st eigenvalue of healthy cell population
- Design of an augmented Lagrangian by combining  and 2-(with penalty)
- Arrow-Hurwitz (or Uzawa) algorithm to track saddle points of the Lagrangian
- …thus obtaining only suboptimality (necessary to obtain critical points) conditions
Circadian + pharmacological control on transitions
K(x,t) = k(x).(t).[1-g(t)]: k FUCCI-identified, clock, g drug effect to be optimised
green and red: ,
gating at transitions
blue: (1-g).
therapy g blocks 
Evolution of the two populations of cancer and healthy cells
Cancer
Circadian control,
no drug infusion
Healthy
Circadian control,
added drug infusion
Healthy
Cancer
(F. Billy et al. 2012)
Numerical solution to the optimal infusion problem
and effect on eigenvalues, healthy and cancer
Infusion scheme g(t)
Target eigenvalues:
Cancer (blue)
Healthy (green)
In favour of this approach:
- characterises long-term
trends with one number,
- easily accessible
target for control
- fits to physiologically
structured growth models
Its drawbacks:
- deals with asymptotics,
not with transients
- assumes a linear model
for proliferation
- assumes periodic control
by drugs (but the period
can be infinitely long)
Now tackling the problem of drug resistance
...within the “DarEvCan” French research consortium
Darwinian Evolution and Cancer
http://www.darevcan.univ-montp2.fr/
A soaring theme on the international scene:
Carlo Maley
Robert Gatenby, MD*
First international
Evolution and cancer conference
SF, June 3-5, 2011, next one in 2013
* RG advocates ‘adaptive therapy’, cf. Gatenby Nature 2009, Gatenby et al. Cancer Research 2009
Gatenby’s new paradigm: rational management of cancer burden by ‘adaptive therapy’
See also review on evolution and cancer by Aktipis et al. PLoS One, Nov. 2011
Tackling this other issue in cancer pharmacotherapeutics:
Emergence of drug resistance in cancer cell populations
Instead of controlling drug resistance at the individual cell level (ABC transporters),
representation of the possible emergence of resistant cell clones due to mutation
occurring at mitoses or adaptation to the environment in a cell Darwinism perspective.
Assumption: cancer cell populations, under the pressure of a drug-enriched
environment, may develop (costly) mutations or adaptations yielding resistant cell
clones, less fit in a drug-free environment, but better survivors in a hostile environment.
A therapeutic objective, under these circumstances, may be not to eradicate all
cancer cells (in fact only all drug-sensitive cells), but instead to let some of them
live so as to limit the growth of an emergent resistant cell clone (‘adaptive therapy’).
Mathematical modelling: design of cell population models structured according to a
phenotypic trait (representing drug resistance, cell division cycle not considered here)
First point of view: ‘mutations only’, one cytotoxic drug
a) Healthy cells
First point of view: ‘mutations only’, one cytotoxic drug
b) Cancer cells
Point of view ‘mutations only’: monomorphism in the healthy cell population
Point of view ‘mutations only’: monomorphism also in the cancer cell population
Simulations by Alexander Lorz (JC, T. Lorenzi, A. Lorz, B. Perthame, in revision)
2nd point of view : two different drugs, cytotoxic and cytostatic, two resistance
traits x and y,‘no mutations, exchanges with the environment instead’
Monomorphism in the healthy cell population
No mutations: non-resistant (‘healthy’) cells: starting from a common medium phenotype
(cytotoxic res.=.5, cytostatic res.= .5), evolution towards the non-resistant (0,0) phenotype
Model, simulations and figures by Tommaso Lorenzi
Dimorphism in the cancer cell population
No mutations: Resistant (‘cancer’) cells: starting from the same common medium phenotype
(.5,.5), evolution towards 2 different resistant phenotypes: (1,0) and (0,1)
Model, simulations and figures by Tommaso Lorenzi
What more now? Need for multiscale models!!
Drugs act at the molecular and cell level, produce measurable effects
at the cell population level, and are delivered at the whole body level
Modelling challenge: integrating
these levels in whole body models
to optimise drug control on tissue
proliferation, healthy and cancer,
in a systems biology perspective:
- intracellular signalling networks,
actual drug targets
- cell populations: proliferation,
apoptosis, differentiation
- whole body ensemble of tissues
as interacting cell populations
under drug control
(JC, Personalized Medicine, 2011)
Collaborators
Bang & LJLL: Annabelle Ballesta, Fadia Bekkal Brikci, Frédérique Billy, Luna Dimitrio,
Marie Doumic, Pierre Gabriel, Herbert Gayrard, Erwan Hingant, Thomas Lepoutre,
Tommaso Lorenzi, Alexander Lorz, Thomas Ouillon, Benoît Perthame, Emilio Seijo Solis
Other INRIA teams: Catherine Bonnet (Disco), Stéphane Gaubert, Olivier Fercoq
(Maxplus), Jean-Charles Gilbert (Estime), Mostafa Adimy, Samuel Bernard, Fabien
Crauste, Stéphane Génieys, Laurent Pujo-Menjouet, Vitaly Volpert (Dracula)
INSERM U776 “Biological Rhythms and Cancers” (Francis Lévi, Paul-Brousse hospital,
Villejuif): Solid tumours of Mice and Men, chronotherapeutics of colorectal cancer
UMRs UPMC- INSERM U872 Team 18 “Resistance and survival of tumour cells”
(J.-P. Marie, Cordeliers Research Centre and St Antoine Hospital, Paris): Leukaemias
Université Paris-Nord (Claude Basdevant): Optimal control theory and algorithms
FP7 ERASysBio C5Sys: http://www.erasysbio.net/index.php?index=272
ANR Bimod (Lyon, Paris, Bordeaux)
GDR DarEvCan (Montpellier, Paris, Lyon,...): http://www.darevcan.univ-montp2.fr/